CMSC 858F: Algorithmic Lower Bounds Fall SAT and NP-Hardness
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1 CMSC 858F: Algorithmic Lower Bounds Fall SAT and NP-Hardness Instructor: Mohammad T. Hajiaghayi September 23, 2014 The most important NP-Complete (logic) problem family! 1 SAT = Satisfiability The Satisfiability problem (introduced by Cook in 1971 [2] and again by Levin in 1973 [11] 1 ), asks the following question: given a boolean formula F (AND, NOT, and OR) over n variables x 1,..., x n, does there exist a set of assignments for each variable x i such that F is TRUE. This problem is well studied and is a well known example of an NP-Complete problem. 2 Circuit SAT The formula to be satisfied can also be expressed as a circuit of logic gates (allows re-use). x 1 x 2 x 3 1 This citation is to a survey in English which translates the original Russian paper. You can find the Russian version here if you re curious. 1
2 3 CNF SAT [2] CNF = Conjunctive Normal Form The formula consists of a conjunction (logical AND) of clauses Each clause is a disjunction (logical OR) of literals Each literal {x i, x i } A CNF formula can also be represented as a bipartite graph. One set contains the variables, the other set contains the clauses, and edges are drawn from variables to the clauses they are in. In this case, there are two types of edges, representing positive and negative literals, respectively. x 1 x 2 x 3 x 4 (x 1 x 2 x 4 ) (x 2 x 3 x 4 ) The first set can contain all of the literals rather than the variables. If so, literals of the same variable (the positive and the negative) must have an edge between them (making this technically not bipartite). x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 (x 1 x 2 x 4 ) (x 2 x 3 x 4 ) Note that if the bipartite graph between clauses and variables is planar; the problem is called planar CNF (type 1). If the bipartite graph between clauses and literals plus all edges (x i, x i ) forms a planar graph, the problem is called planar CNF (type 2). Both versions are NP-Complete by a reduction from 3-SAT (by uncrossing the edge-crosses). Very useful to prove NP-Hardness of problems on planar graphs and geometric planar graphs SAT [2] Each clause is an OR of exactly 3 variables, i.e., the degree of each clause is 3. 2
3 SAT-5 [5] Each variable occurs in 5 clauses Monotone 3-SAT [7] Each clause consists of either 3 positive literals or 3 negative literals. There is no mixing of signs. 4 Polynomial-Time Variants of SAT : All of the above variants of SAT are NP-Complete, but beware of the polynomialtime variants below! SAT Each clause is the logical OR of two literals The satisfiability of a formula in this form can be determined in polynomial time First, notice that each clause is of the form (x y), but this is equivalent to (x = y) or (y = x) This causes implication chains where we can guess a value for x, forcing assignments for the rest of the values. To solve in polynomial time, first try TRUE as an assignment for x and if the formula is satisfied, you re work is done. If not, try FALSE. If the formula is still not satisfied, then it must not be satisfiable. BUT MAX 2-SAT [6] P Q P = Q T T T T F F F T T F F T A simple modification to the 2-SAT problem will make it NP-Complete. The MAX 2-SAT problem asks for a satisfying assignment of the variables that maximizes the number of TRUE clauses. As always, we can turn an optimization problem into a decision problem by assigning a bounding variable (e.g., can we make more than K clauses TRUE?). 3
4 4.2 Horn SAT A Horn SAT formula is one in which each clause has 1 positive literals. For example, a Horn Clause may take the form x y z w. However, thanks to DeMorgan s Law, this is equivalent to (x y z) w. This, in turn, is equivalent to (x y z) = w, leading to an implication change, making this a generalization of 2-SAT and thus solvable in polynomial time. 4.3 Dual Horn SAT In Dual Horn SAT, each clause has 1 negative literal. This can be reduced to Horn SAT by negating everything (both the literals and the final variable assignment), making it solvable in polynomial time. 5 DNF DNF = Disjunctive Normal Form The formula consists of a disjunction (logical OR) of clauses Each clause is a conjunction (logical AND) of literals Each literal {x i, x i } To satisfy a formula it is sufficient to satisfy a single clause, which is done as long as there are no contradictions, i.e., a clause cannot include both the positive and negative literals of a variable. Thus, it is essentially trivial. Attempting to make the formula FALSE instead is, essentially, the dual of a CNF SAT problem. 6 Alternative Clauses for 3-SAT in-3-SAT [10] Exactly 1 of 3 literals in a clause is TRUE Monotone 1-in-3-SAT Similar to 1-in-3-SAT, though no negations are allowed, i.e., all literals are positive. This formulation was omitted by Schaefer. BUT... 4
5 6.1.2 Monotone Not Exactly 1-in-3-SAT [10] Again, all literals are positive. Not Exactly means that only 0, 2, or 3 variables in a clause may be TRUE I.e., x i = (x j x k ) Dual Horn SAT Additionally, x 1 must be assigned the value TRUE (otherwise you could just make everything FALSE) Polynomial 6.2 NAE 3-SAT [10] NAE = Not All Equal The 3 literals in a clause may not all be assigned the same value I.e., FFF and TTT are forbidden assignments in a clause, whereas only FFF is forbidden in 3-SAT this leads to a nice symmetry between TRUE and FALSE Monotone NAE 3-SAT No negations, all literals are positive. Also omitted by Schaefer. Also NP-Complete Of all of the problems listed above, the most important ones to remember are: 3-SAT, 1-in-3-SAT, and NAE 3-Sat. 7 Shaefer s Dichotomy Theorem [10] (A Universal Theorem) A formula = AND of clauses general clause = relation on variables (with implicit truth tables, similar to a type in C++ or having an oracle). assume in CNF = AND of subclauses SAT is polynomial if either: 5
6 setting all variables to TRUE or all variables to FALSE satisfies all relations, OR - subclauses are all Horn or all Dual Horn, OR - relations are all 2-CNF (subclause sizes 2, i.e., 2-SAT case), OR - every relation can be expressed as a system of linear equations over Z 2. x 1 x 2 x 3 x 4 = 0or1 where = XOR. This can be solved via Gaussian elimination. If none of the above are true, then SAT is NP-Complete! 8 2-Colorable Perfect Matching [10] given a planar 3-regular graph (i.e., a graph in which the degree of each vertex is 3). 2-color the vertices such that every vertex has exactly 1 same-colored neighbor. special case of 2-in-4-SAT Red Red Red Edge (planarity and 3-regular are left as an exercise) Theorem 1 There is a reduction from Monotone NAE 3-Sat to 2-Colorable Perfect Matching [10]. 6
7 Proof: The proof is based on two key gadgets, as illustrated in the image on the right, taken directly from the paper by Schaefer [10]. Figure 1(a) is a gadget used to represents the individual clauses in a 3-SAT formula. A vertex is created for each literal, which are all linked with an edge. Additionally, an intermediate vertex is added in the center to which each variable is linked. In this way, one cannot set each variable to the same color. The second gadget, illustrated in Figure 1(b), is a copy gadget. In this case, the color of n 1 must be equal to n 2. Finally, figure 1(c) illustrates the reduction of A to a 2-Colorable Perfect Matching problem. Some fun games can be proved to be NP-Complete via reduction from 3-SAT, e.g. Push-1 [8]. Image of a Push-1 (Sokoban) game taken from Wikipedia - Sokoban. You may see Erik s class for high-level ideas. The hardness of other games can proven as well, such as: Super Mario Bros. Legend of Zelda Metroid Donkey Kong Country Pokemon 7
8 9 Cryptarithms/Alphametics [9] given a formula x + y = z with each number written in base b (which should be some f(n) to be interesting) and encoded with letters by an unknown bijection between {0, 1,..., b 1} and letters goal: feasable? / recover bijection strongly NP-Complete [3]. Cryptarithm rules: each letter represents a unique digit often numbers must not start with zero often the solution is unique Example: Can also be represented as: Or: a b c d + e f g b e f c b h S E N D + M O R E M O N E Y 9.1 Reduction from 3-SAT To reduce from 3-SAT, a long equation is created that represents the clauses, variables, and the boolean nature of the SAT problem. First, the rightmost three columns are set thusly: 0 p 0 0 p 0 1 q 0 Here the letters 0 and 1 are forced to be 0 and 1 for any mod. Next, for each pair of literals v i and v i, the portion below is added to the left side of the formula: 0 d i 0 1 y i 0 c i y i 0 b i y i 0 a i 0 0 e i 0 d i y i 0 c i y i 0 b i y i 0 a i 0 v i 0 e i z i 0 d i z i 0 v i z i 0 b i a i 0 8
9 Thus, for the variable gadget: b i = 2a i v i = 2b i + C (where C = carry(y i + y i ) {0, 1}) = 4a i + C C(mod4) d i = 2c i + C e i = d i C = 2c i C v i = d i + e i = 4c i C Next, for each clause add: 3C C(mod4) Thus, for the clause gadget: 0 u ab 0 v a 0 1 r i 0 g i w i 0 f i 0 0 v c 0 v b 0 h i r i 0 g i w i 0 f i 0 0 t i 0 u ab 0 t i s i 0 h i x i 0 g i 0 g i = 2f i h i = 2g i + {0, 1} = 4f i + {0, 1} t i = h i {0, 1} = 4f i {0, 1, 2} = 4f i + {1, 2, 3} v a + v b + v c = t i {1, 2, 3}(mod4) The reduction is good for NP-Completeness for any mod multiple of 4, but we still need a solution for the puzzle from a satisfying solution, e.g. uniqueness issues. 9.2 Simplified Reduction From 1-in-3-SAT Variable gadget: just v i, no v i (monotone) Clause gadget: g i = 2f i h i = 2g i = 4f i t i = h i + 1 = 4f i + 1 v a + v b + v c = t i = 4f i + 1 1(mod4) 9
10 9.3 3-SAT Solvable = Cryptorithm Solvable distinguish a i, b i, c i, d i,... by value mod 128 (one class for each variable) e.g., below are possible choices for each variable: v i 8(mod128) if TRUE 9(mod128) if FALSE a i {2, 34, 66, 98}(mod128) b i {4, 68} set v i 128 and vi 128 [0, (2n) 3 ] such that we have distinct sums of triples for all clauses It was proven that for any k there is a set of k numbers all between 1... k 3 such that their sum in triples are distinct (we can use powers of 2 but the base would be in O(4 n ) and not good for strong NP- Hardness)[1]. easy proof of polynomial range (based on fusion trees): if < i set by induction, v i must avoid v j + v k v l v m v p < (2n) 5 choices = (2n) 5 suffices = strongly NP-hard The final result would be in base (2n) = 3072n 3 (see [3] and its revision [4] for all of the details). A Notes It is unknown whether or not the graph isomorphism problem is NP-Complete. Because of this, researchers have defined a new class GI, the set of problems which are polynomial-time reducible to the graph isomorphism problem. If the graph isomorphism problem is solvable in polynomial time, then GI would be equal to P. References [1] SC Bose and S Chowla. Report inst. theory of numbers, [2] Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC 71, pages , New York, NY, USA, ACM. 10
11 [3] D Eppstein. On the np-completeness of cryptarithms. ACM SIGACT News, 18(3):38 40, [4] David Eppstein. On the np-completeness of cryptarithms. Computer Science Department, Columbia University, URL: ics. uci. edu/ eppstein/pubs/epp-sn-87. pdf, pages 06 8, [5] Uriel Feige. A threshold of ln n for approximating set cover. JOURNAL OF THE ACM, 45: , [6] M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NPcomplete graph problems. Theoretical Computer Science, 1(3): , February [7] E Mark Gold. Complexity of automaton identification from given data. Information and Control, 37(3): , June [8] Michael Hoffmann. Motion planning amidst movable square blocks: Push-* is np-hard, [9] Joseph S Madachy. Madachy s Mathematical Recreations. Dover Publications, [10] Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 78, pages , New York, NY, USA, ACM. [11] B.A Trakhtenbrot. A survey of russian approaches to perebor (brute-force searches) algorithms. Annals of the History of Computing, 6(4): , October
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